Estimating Spillovers using Panel Data, with an Application to

the Classroom∗

Peter Arcidiacono, Duke University

Gigi Foster, University of South Australia

Natalie Goodpaster, Analysis Group, Inc.

Josh Kinsler, University of Rochester

February 12, 2009

Abstract

Obtaining consistent estimates of spillovers in an educational context is hampered

by two issues: selection into peer groups and peer effects emanating from unobservable

characteristics. We develop an algorithm for estimating spillovers using panel data that

addresses both of these problems. The key innovation is to allow the spillover to operate

through the fixed effects of a student’s peers. The only data requirements are multiple

outcomes per student and heterogeneity in the peer group over time. We first show that the

non-linear least squares estimate of the spillover is consistent and asymptotically normal

as N →∞ with T fixed. We then provide an iterative estimation algorithm that is easy to

implement and that converges to the non-linear least squares solution. Using University

of Maryland transcript data, we find statistically significant peer effects on course grades,

particularly in courses of a collaborative nature. We compare our method with traditional

approaches to the estimation of peer effects, and quantify separately the biases associated

with selection and spillovers through peer unobservables.∗We thank Joe Altonji, Pat Bayer, Paul Frijters, John Haltiwanger, Caroline Hoxby, Tom Nechyba, Bruce

Sacerdote, seminar participants at Duke University, University of Georgia, Yale University, and participants at

the 2004 Labour Econometrics Workshop held at the University of Auckland and the NBER’s Higher Education

Workshop for helpful comments, as well as the University of Maryland administrators who have provided us

with the data used in this paper. Shakeeb Khan was particularly helpful. Special thanks go to Chris Giordano

and Bill Spann of the Maryland Office of Institutional Research and Planning. We also thank Graham Gower

for superior research assistance.

1

1 Introduction

The question of how peers affect student achievement underlies many debates in applied

economics. Peer effects are relevant to the estimation of the impact of affirmative action,

school quality, and public school improvement initiatives such as school vouchers, and are

central to more immediate concerns such as how best to group students to maximize learning.

However, despite this wide field of potential relevance, the empirical estimation of spillovers—

whether in the education context or elsewhere— is not straightforward.

There are at least two barriers that must be overcome when estimating spillovers on stu-

dent achievement. The first is the selection problem. When individuals choose their peer

groups, high ability1 students may sort themselves into peer groups with other high ability

students. With ability only partially observable, positive estimates of peer effects may result

even when no peer effects are present because of a positive correlation between the student’s

unobserved ability and the observed ability of his peers. Researchers have undertaken a variety

of estimation strategies to try to overcome the selection problem,2 but significant empirical

problems linger, both because researchers only have access to incomplete measures of ability

and because peer effects may operate differently when peers are chosen rather than assigned.

A second barrier is that spillovers may work in part through characteristics that are not

observed by the econometrician. The importance of peer effects may be significantly under-

stated if the primary channel through which they operate is unobserved. Peer effects through

unobservables has received little attention outside of Altonji et al. (2004). Random assign-

ment is able to circumvent the selection problem but the obstacle to estimation posed by peer

effects through unobservables remains.

We introduce a new algorithm for estimating spillovers using panel data that overcomes

both these obstacles. The key innovation is that the peer effects are captured through a linear

combination of individual fixed effects. Additional types of fixed effects, such as institutional1For ease of exposition we refer to the bundle of individuals’ performance-relevant characteristics as ‘ability.’2One set of papers uses proxy variables to break the link between unobserved and peer ability (Arcidiacono

and Nicholson (2005), Hanushek et al. (2003), and Betts and Morell (1999)). Another set of papers relies on

some form of random assignment (Sacerdote (2001), Zimmerman (2003), Winston and Zimmerman (2003),

Foster (2006), Lehrer and Ding (2007), and Hoxby (2001)). Finally, researchers have tried to circumvent the

endogeneity problem with instrumental variables (Evans et al. (1992)).

2

effects, can also be incorporated into the algorithm. Thus our method, while originally devel-

oped for use in social effects research, can also be used in applications where fixed effects of

large dimension and of potentially multiple types must be estimated.

Constructing the spillover as a linear combination of individual fixed effects results in

a non-linear optimization problem. Estimating individual unobserved heterogeneity in non-

linear panel data models often results in biased estimates of the key parameters of interest—

the incidental parameters problem.3 As N goes to infinity for a fixed T , the estimation error

for the individual effects often does not vanish as the sample size grows, contaminating the

estimates of the parameters of interest.4 We show, however, that the nonlinear least squares

estimate of the spillover parameter is consistent and asymptotically normal as N → ∞ with

T fixed, even though the fixed effects themselves are not consistent.5

While nonlinear least squares yields consistent estimates of the spillover parameters, the

dimensionality of the problem renders nonlinear least squares infeasible. We develop an it-

erative algorithm that, under certain conditions, produces the same estimates as nonlinear

least squares. The algorithm toggles between estimating the individual fixed effects and the

spillover parameters. Each iteration lowers the sum of squared errors, with a fixed point

reached at the nonlinear least squares solution to the full problem.

As a precursor to the exposition of our full spillover model, we show how our iterative

estimator can be used to identify multiple types of fixed effects in contexts where no spillovers

are being estimated. Then, we customize the model for application to peer effects in education,

and estimate the customized model using student-level data from the University of Maryland.

Six semesters of transcript data are available covering the semesters from the spring of 1999

to the fall of 2001. We observe grades for every class each student took over the course of this

period as long as the student lived on campus during any one of the six semesters.

We estimate the model separately for each of three types of courses where ability is allowed3Neyman and Scott (1948) were the first to document the incidental parameters problem.4Hahn and Newey (2004) provide two methods of bias correction: a panel jackknife and an analytical correc-

tion. Woutersen (2002) and Fernandez-Val (2005) consider estimators from bias-corrected moment conditions.

In a similar vein, Arellano and Hahn (2006) and Bester and Hansen (2005) consider bias-correcting the initial

objective function.5Other special cases where the incidental parameters problem does not require a bias correction are Manski

(1987), Honore (1992), and Horowitz and Lee (2004).

3

to vary by course type. We find significant peer effects which vary by course type. A one

standard deviation increase in peer ability yields average returns similar to those from between

a 3 percent and a 11 percent of a standard deviation increase in own ability, depending upon

the course type and specification. The lowest returns are found in math and science with the

highest returns found in the social sciences.

Our model allows us to quantify selection both within and across course types. Within

course types, we compare the amount of selection with respect to observed and unobserved

student ability. To arrive at these measures, we decompose each of the estimated student fixed

effects, or what we label total ability, into an observed and an unobserved component using

typical observed ability measures, such as SAT scores and high school performance. For all

course types we find greater selection on unobserved ability than observed ability. However,

selection is highest when measured using total ability, a reflection of the significant correlation

between peer observed and unobserved ability within a section.6 Because we estimate our

model separately for each course type, we can compare student ability in the primary course

of study to ability in other fields, thereby quantifying selection across course types. We find

strong evidence of both comparative and absolute advantage. On average students select

course types for which they are best suited. However, math and science students show greater

aptitude overall in every course type.

Finally, we compare our peer effect estimates to those that would be obtained using more

conventional methods. In particular, we examine separately the two obstacles present in

traditional peer effects estimation: selection into peer groups and the effect of peer unobserv-

ables. Controlling for selection only, which is what is accomplished using random assignment,

we show that the estimated peer effects are lower than the peer effects obtained using our

method. This is because random assignment techniques rely on incomplete measures of peer

ability. We then re-introduce selection into the model and show that the bias in the peer

effect estimate can be either positive or negative when both issues are present. For humanities

courses, the peer effect estimate continues to be biased downwards since the peer unobserv-

ables problem dominates the selection issue. The opposite is true for math and science courses

where the peer effect estimate using conventional methods is more than four times our original

estimate. The differences across course types is due in part to the much higher correlation6By construction, observed and unobserved ability are orthogonal in the population.

4

between individual observed ability and peer unobserved ability in math and science relative

to the humanities.

The remainder of the paper proceeds as follows. Section 2 discusses our iterative estima-

tion algorithm in the context of some outstanding problems in the applied microeconomics

literature. Section 3 presents the model, its properties, our estimation strategy, and evidence

from Monte Carlo simulations. Section 4 describes the University of Maryland data, and

Section 5 presents the results. Section 6 explores selection within and across course types

and Section 7 illustrates the biases associated with traditional peer effect measures. Section

8 concludes.

2 Estimating Models with Multiple Types of Fixed Effects

An outstanding problem in the applied microeconomic literature is how to estimate models

containing multiple types of fixed effects where each set of fixed effects is of a large dimension.

We begin with this econometric problem since the iterative method we employ solves the issue

of multiple fixed effects en route to estimating spillovers. We then present our full spillover

model in the next section. We focus on two papers in particular: Rivkin et al. (2005) and

Abowd et al. (1999), to illustrate the difficulties in estimating large numbers of fixed effects.

Rivkin et al. (2005) model gains in test scores as a function of the observed characteristics

of the students, Xi, and teacher fixed effects, δj , where i indexes individuals and j indexes

teachers. The change in test scores from time t− 1 to t given that the individual has teacher

j at time t, ∆Y, is then modeled as:

∆Y = αXi + δj + εit (1)

Note that Xi includes characteristics of the students that do not vary over time. However,

Xi may not include the full set of individual characteristics that are relevant for achievement

gains, and the omitted variables may be correlated with the δj ’s due to streaming of students

and/or systematic selection of certain teachers into classrooms with higher- or lower-ability

students. As an alternative, we could estimate the model with both student and teacher fixed

effects:

∆Y = αi + δj + εit (2)

5

However, estimating both sets of fixed effects simultaneously would be infeasible given the

large number of students and teachers in their data.

Abowd et al. (1999) are interested in estimating models that include both firm and worker

fixed effects.The most basic model they are interested in estimating contains just individual

and firm-specific effects regressed on log earnings. Labeling log earnings for individual i in

firm j at time t as Y, the outcome equation is:

Y = αi + δj + εit (3)

A more interesting case occurs when there are tenure effects that vary across firms. For

simplicity, assume that the effects of tenure are linear. Labeling X as the amount of tenure

individual i has in firm j at time t, the outcome equation is:

Y = αi + δj + φjX + εit (4)

where φj is the firm-specific return to tenure. Abowd and Kramarz (1999) recognize that with

over 1 million workers and 500,000 firms, they cannot estimate the above equation directly.

Instead, they consider a number of estimation techniques, none of which results in least squares

estimates of the firm and worker fixed effects without imposing additional assumptions on the

data generating process.

Our approach yields least squares estimates of both firm and worker effects in a compu-

tationally feasible way without imposing any extraneous orthogonality conditions. Consider

the following regression equation which nests the models from both Rivkin et al. (2005) and

Abowd and Kramarz (1999):

Yijt = αi + βXit + δj + γjZ + εijt (5)

Estimating this equation by OLS solves:

minα,γ

N∑

i=1

T∑

t=1

(Yijt − αi − βXit − δj − γjZ)2 (6)

Minimizing this function in one step remains infeasible as a result of the large number of

firms and workers. Instead, we propose an iterative method that yields OLS estimates of the

parameters of interest while circumventing the dimensionality problem. Given starting values

for the α’s and the β’s, the algorithm iterates on two steps with the qth iteration following:

6

• Step 1: Conditional on αq−1 and βq−1, estimate δq and γq by OLS.

• Step 2: Conditional on δq and γq, estimate αq and βq by OLS.

The process continues until the parameters converge. Because the sum of squared errors is

decreased at each step, we will eventually converge to the parameter values that minimize the

least squares problem in (6), regardless of which pair of parameters we guess first to start the

algorithm. The primary advantage of our method in applications such as those described in

this section is that it is capable of estimating extremely large sets of fixed effects in a reasonable

amount of time. The model becomes significantly more complicated when the outcomes are

allowed to depend on functions of the fixed effects themselves. We discuss this enhancement

now.

3 Estimating Spillovers with Panel Data

In this section we present a model and estimation strategy for measuring achievement spillovers

using student fixed effects, building upon the iterative technique outlined in the prior section.

The model is constructed keeping in mind that our application will be measuring peer effects

in college, where we are interested in the interactions that occur within discussion sections in

large classes.

We first consider a case where one’s outcome depends only on one’s own fixed effect as well

as the fixed effects of the other individuals in a defined peer group. We show that it is possible

to obtain consistent estimates of the spillover and that there is a computationally cheap way of

obtaining the solution. Although we focus on the case where the spillover effect is homogeneous

across individuals, all proofs go through when the spillover is allowed to vary provided the

number of parameters governing the spillover effect does not vary with the sample size. We

then show how the model is easily expanded to incorporate correlated effects, highlighting

again the advantages of iterative estimation in solving the dimensionality problem. Finally,

we provide Monte Carlo evidence that the small-sample behavior of our iterative estimator is

also appealing. All proofs are in the appendix.

7

3.1 Baseline Model

Our baseline model has individual i’s outcome at time t, Yit, depending upon his own observed

and unobserved permanent characteristics, Xi and ui, the observed and unobserved permanent

characteristics of each of the other students in his peer group, and a transitory error εit. Define

Iijt as an indicator function that takes on a value of one if individuals i and j are in the

same peer group at time t, and zero otherwise, and let N be the number of students in the

population. Our baseline specification is then:

Yit = Xiβ1 + uiβ2 +N∑

j 6=i

Iijt (Xjγ1 + ujγ2) + εit (7)

The specification in (7) is restrictive along a number of dimensions. In the language of

Manski (1995), there are no endogenous effects as Yjt has no direct effect on Yit for any

j, and there are no correlated effects as there are no variables to capture the commonality

of the environment faced by all members of student i’s time-t peer group. Further, there

are no accumulation effects as spillovers today do not affect outcomes tomorrow. However,

even in this special case estimation is still problematic when peer groups are chosen. In

particular, there may be correlation between ui and the sum of observed peer characteristics,

leading to biased estimates of γ1. Also, we will not be able to capture the peer influence

through unobservables, meaning that γ2 is inestimable without further assumptions. While

random assignment can remove the correlation between ui and observed peer characteristics,

the inability to capture spillovers through unobservables remains.7

We next make one additional assumption: the relevance to outcomes of peer characteristics

is proportional to that of own characteristics, meaning that we can write γ1 and γ2 as

γ1 = γβ1

γ2 = γβ2

This implies, for example, that if two dimensions of an individual’s ability are equally im-

portant in their effect on Yit, then those two dimensions of peer ability will also be equally

important in determining Yit. This same assumption is used in Altonji et al. (2004).7Using random assignment to identify the spillover also disregards the possibility that spillovers operate

differently in selected versus randomized contexts.

8

Now define αi as:

αi = Xiβ1 + uiβ2

We can then rewrite equation (7) as:

Yit = αi +N∑

j 6=i

Iijt(γαj) + εit (8)

An individual’s outcome is then a function of the individual’s fixed effect plus a linear function

of the fixed effects of the other students in the peer group. Using fixed effects in this way

allows us to abstract from many other covariates that may affect student outcomes. All of

the heterogeneity in fixed student characteristics that might affect student outcomes, such as

birth cohort, sex, IQ, or race is captured with this one measure.

Our goal is then to show the properties of the solution to the non-linear least squares

problem:

minα,γ

N∑

i=1

T∑

t=1

Yit − αi −

N∑

j 6=i

Iijt(γαj)

2

(9)

Given the non-linearities present in the problem, one may suspect that for fixed T , the es-

timates of γ’s will be biased as a result of the incidental parameters problem. However, we

show that under standard assumptions this is not the case.

Theorem 1. Under the following assumptions, the estimates of γ from solving the minimiza-

tion problem in (9) are√N consistent and asymptotically normal for fixed T as N → ∞

where N is the number of peer groups:

1. εit⊥εjs ∀ j 6= i, t 6= s

2. εit⊥αj ∀ i, j

3. Var(εit) is finite and constant across i and t

4. Var(αi) is finite

The structure of the proof relies on solving each of the α’s as a function of the data and γ

and then substituting these functions in for the α’s in (9). Minimizing with respect to γ alone

then yields the result.

9

While concentrating out the α’s is useful for proving consistency, the formulas are quite

cumbersome and difficult to calculate. Directly solving (9) is also generally not possible

because of the dimensionality of the problem. Instead, we consider an iterative estimation

strategy that both circumvents the dimensionality problem and yields the same solution as

direct maximization.

To implement the algorithm, consider the first order condition of the nonlinear least squares

problem with respect to αi:

0 =T∑

t=1

Yit − αi −

N∑

j 6=i

Iijt(γαj)

+

N∑

j 6=i

Iijtγ

Yjt − αj −

K∑

k 6=j

Ijkt(γαk)

(10)

Solving for αi yields:

αi =

∑t

[Yit −

∑j 6=i Iijtγαj +

∑j 6=i γ

(Yjt − αj −

∑k 6=i,j Ijktγαk

)]

T +∑

t

∑j 6=i Iijtγ2

(11)

Note that we have pulled out the αi terms from the last term in (10) to derive (11). We

establish in Theorem 2 the conditions under which, given any initial set of α’s, repeatedly

updating the α’s using (11) yields a fixed point.

Theorem 2. Denote f(α) as a function mapping from RN → RN where the ith element of

f(α) is given by the right hand side of (11) ∀i ∈ N . A sufficient condition for f(α) to be a

contraction mapping is that the maximum of γ/M is less than 0.4, where M is the number of

other individuals in each student’s peer group.

The restriction on the maximum value γ is needed to ensure that the feedback effects are

not too strong. With Theorem 2 giving a solution method for the α’s conditional on the γ’s,

our algorithm iterates on estimating the α’s using f(α) (taking the γ’s as given), and then

estimating the γ’s taking the α’s as given. Each of these two steps lowers the sum of squared

errors and, analogous to the estimator in Section 2, converges to the nonlinear least squares

solution. In practice, we have found that the algorithm performs substantially faster if the

α’s are only updated until the sum of squared errors falls before moving on to re-estimating

γ.8 To summarize, the algorithm is started with an initial guess for the α’s and iterates on

two steps until convergence, with the qth iteration given by:8For most iterations of our models updating the α’s just once led to a decrease in the sum of squared errors.

10

• Step 1: Conditional on αq−1, estimate γq by OLS.

• Step 2: Conditional on γq, update αq according to (11).

3.2 Correlated Effects

We now extend the model to allow for correlated effects. If each individual peer group is

exposed to a different environment, it is impossible to separate the correlated effects from the

exogenous effects without further parameterizations. A restriction that is easily imposed for

correlated effects in our context is a component to one’s grade that is course-specific. This is

important as grade inflation and where the curve lies will affect one’s final grade irrespective

of own and peer ability. Denote Iict as an indicator for whether individual i is in course c at

time t where c ∈ [1, C]. Placing course fixed effects into the achievement equation yields:

Yit = αi +N∑

j 6=i

Iijt(γαj) +C∑

c=1

Iictδc + εit (12)

The δc’s are then the course fixed effects used to capture correlated effects since all peer groups

are formed within courses.

The non-linear least squares problem we are now interested in solving takes the following

form:

minα,γ,δ

N∑

i

T∑

t=1

Yit − αi −

N∑

j 6=i

Iijt(γαj)−C∑

c=1

Iictδc

2

(13)

While the consistency of γ is unchanged regardless of whether we include other time-varying

regressors, it is particularly clear here since we can re-write the above expression without δc

by demeaning the dependent variable at the course level.

The first order condition with respect to αi changes to reflect the presence of the course

fixed effects:

αi =

∑t

[Yit −

∑c Iictδc −

∑j 6=i γαj +

∑j 6=i Iijtγ

(Yjt − αj −

∑c Iictδc −

∑k 6=i,j Ijktγαk

)]

T +∑

t

∑j 6=i Iijtγ2

(14)

The updating rule for the α’s then follows directly from (14), once α terms are collected for

each student.

For a given set of α’s and γ’s, the course fixed effects can be calculated according to:

δc =

∑i

∑t Iict

(Yit − αi −

∑j 6=i Iijtγαj

)∑

i

∑t Iict

(15)

11

The estimation strategy is then the same as without correlated effects, with one additional

step. As before, each step of the estimation decreases the sum of squared errors. The algorithm

is set by an initial guess of the α’s and δ’s and then iterates, with the qth iteration following:

• Step 1: Conditional on αq−1 and δq−1, estimate γq by OLS.

• Step 2: Conditional on δq−1 and γq, update αq according to (14).

• Step 3: Conditional on αq and γq, update δq according to (15).

Adding other types of fixed effects simply adds additional steps to the estimation, with each

set of fixed effects updated in an additional step. For example, had data been available on

teaching assistants, it would have been possible for us to control for teaching assistant effects

as well.

3.3 Monte Carlo Simulations

To investigate the properties of our iterative estimator in the presence of correlated effects, we

now run simulations using different assumptions about the empirical setting. In each setting,

the model is simulated using 10,000 students. We simulate the model 100 times under various

states of the world constructed by varying four dimensions of the problem:

1. Observations per student- The number of outcomes observed per student varies across

simulations between 2, 5, and 10. 5 is the maximum number of observations a researcher

may have when analyzing grade school or high school test score data, and 10 observa-

tions is more likely when analyzing grades achieved in university-level courses. More

observations per student implies more accurate measurement of the α’s.

2. Students per peer group- The number of students per peer group varies across simulations

between 2 and 10. 2 is the minimum number of observations required to identify a

spillover in this type of model. 10 students per peer group is in the range of what might

be observed in typical classroom-based data sets.

3. Selection into classes- To show that our estimator solves the selection problem, we

simulate the model under alternative assignment rules. Under random assignment, the

12

average standard deviation of the α’s within a peer group equals the standard deviation

of α in the population. We also simulate the model with selection such that the average

standard deviation of the α’s within a peer group is 75% of the population standard

deviation. The selection level in the non-random assignment simulation corresponds

with the sorting observed in the Maryland transcript data.9

4. Transitory component- The noisier the outcome measure, the noisier the estimates of

the α’s will be. The distribution of the α’s is set at N(0,1). The ε’s are distributed with

mean zero and standard deviation equal to 1.15 or 1.95.

The common group-level shock used to model the correlated effect is not statistically associated

with the abilities of students in the classes. However, students are sorted into classes based

upon ability. Thus, the average standard deviation of abilities within a class is smaller than

standard deviation of abilities in the population.

To verify first that the estimator does not give positive peer effects when no peer effects

are present, Table 1 shows simulation results when the true peer effect, γ, is zero. In all cases,

regardless of the number of observations per student, the level of selection, and the noisiness

of the outcome measure, γ̂ is precisely estimated around zero. There is no evidence of bias

when the true peer effect is zero.

Table 2 documents the model’s performance when the true value of γ is 0.15. Again,

regardless of assignment procedure or section size, γ is centered around the truth. However,

two interesting patterns emerge in the estimates and standard errors of γ. First, γ is more

precisely measured when students are randomly assigned to sections. Selection in this case

can be thought of as occurring at two levels: the class level and the section level. These

results reflect the fact that selection at the class level confounds the estimate of the correlated

effect and reduces the precision of the section-level peer effect estimate. In fact, if selection

occurred only at the section level, the peer effect estimates would be more precise than in the

random assignment case (ceteris paribus), since there would be greater variation in peer ability.

Second, as the peer group size increases, the precision of γ decreases.10 This is again related9In our data, the standard deviation of ability within peer groups is between 70% and 77% of the standard

deviation of ability in the population depending upon the course type.10This can be seen in Table 2 since as the number of observations per student increases, we should naturally

see an increase in precision—yet we do not, because peer group size increases as well, driving standard errors

13

to the variation in peer ability across sections. With smaller section sizes, other things equal,

there is greater variation in peer ability across sections which yields more precise estimates of

the spillover.

4 Data

With the model producing consistent estimates of the spillover parameter and performing well

in our Monte Carlos, we now turn to the data used in estimation. The administrative data

set used in this paper covers all undergraduates observed residing in University of Maryland

on-campus housing during any of the following six academic semesters: Spring 1999, Fall

1999, Spring 2000, Fall 2000, Spring 2001, or Fall 2001. The data set includes students living

off-campus in a given semester as long as they were observed living on campus during at least

one of the six semesters. 90% of University of Maryland entering freshmen live on campus

in their first semester,11 so the data set includes at least 90% of the University of Maryland

undergraduate population who began study sometime in the six-semester period. There is a

less complete representation for upperclassmen, some of whom entered before our observation

period and may not have lived on campus during the period. However, our identification of peer

effects comes from large, multi-section courses in which freshmen predominate.12 A “section”

in the American undergraduate context is a subset of students from an entire course that

meets together formally at least once a week. In these smaller groups, greater communication

and interaction is expected of students. Our section-level spillover captures how the ability of

other students in the same section impacts any given student’s eventual grade in a course.

To generate the student-section-level sample used to estimate our models, we first placed

two major restrictions on the data set: students had to have valid A through F grade informa-

tion for the given section, and they could not be the only student observed in the section that

semester.13 These restrictions yielded a sample of 300,640 student-section observations. We

up. The negative association of peer group size and precision, as well as all other relationships discussed here,

have also been verified in numerous additional Monte Carlo exercises; results are available upon request from

the authors.11This number is taken from publicly-available statistics posted on the University’s web page.12In tests of whether classes that were under-represented had lower estimated peer effects than those with a

complete representation, we found no meaningful differences.13Numeric grade equivalents were assigned as follows: A = 4; B = 3; C = 2; D = 1; and F = 0. Students who

14

then deleted all observations on sections that were not in one of three well-defined academic

subgroups: (1) humanities (86,844 observations), (2) social sciences (77,312 observations), and

(3) hard sciences and mathematics (82,675 observations).14 This left a combined sample of

246,831 student-section observations, representing 18,511 individual students. Sample sizes

are provided in Table 3.

Finally, while our method does not require the presence of observable characteristics about

individuals, the data set to which we apply it does offer an array of observable measures about

each student. Later in the paper, we use the following information about students in further

analysis of our models’ results: SAT math and SAT verbal scores; high school grade point

average; sex; race; whether the student was included in the honors program; whether the

student participated in sports; and whether the student was an in-state student (as opposed

to out-of-state or international).

5 Estimates of Classroom Spillovers

We now turn to our model specifications and estimates. We first describe and estimate a model

that restricts the peer effect such that the spillover only depends upon the mean ability in the

section. The second specification allows the size of the spillover to depend upon one’s own

characteristics. For example, those with high SAT verbal scores may receive higher benefits

from their peers than those with low SAT verbal scores. With the results of the two models in

hand, we then show how predictable ability is given observable measures such as SAT scores,

high school grade point average, and demographics.

withdrew from a course, audited, or received a non-letter grade (such as Pass) were excluded from the sample

due to concerns that they might not have been present during sections and classes. If two separate grades were

recorded for the student for a given section, the highest grade was used.14Excluded courses include those that are generally more vocationally-oriented, but very diverse; for example,

journalism, nutrition and food science, landscape architecture, and library science. Because our model estimates

a homogeneous underlying ability for each course type, we did not include these courses in a separate category

due to our concern that the underlying ability necessary to succeed in them is not sufficiently homogeneous

across the category.

15

5.1 Homogeneous Gamma Model

With c and t indexing courses and sections, we have the same specification as in equation

(12) except that now, with the number of individuals in each section varying, we restrict the

spillover to depend upon the mean fixed effect of the other individuals in the same section of

a course:

Yit = αi +N∑

j 6=i

Iijt

(γαj

Mit

)+

C∑

c=1

Iictδc + εit (16)

Because grades are assigned at the course level, there is a relationship between students who

share a course but are not in the same section that cannot be captured by the section peer

effect, γ. We might expect, for example, that if the course is graded on a curve and the

entire class is extremely able, a mediocre student’s grade may suffer. Similarly, the teacher

of a course may respond to higher average class ability by teaching in a way that enhances

learning and therefore raises grades for at least some portion of the class. By including fixed

effects at the course level, the δc’s, we can make the outcome measure comparable across

classes.

Consistent with the data section, we split courses into three types: humanities, social

sciences, and math and science. A student’s performance in each type of course will differ

according to the particular student’s strengths and weaknesses. Therefore, instead of encap-

sulating all the attributes of a student into one ability measure, we allow students to have

separate ability measures for each course type in which they are enrolled. As noted above, all

courses used in our analysis are classified as belonging to one of the following course types:

humanities, social sciences, or math and science. We estimate an independent ability measure

for each type of course for each student, conditional on the student’s enrollment in at least

one class within that course type.15 Another supporting rationale for the empirical division

into course types is that the amount of interaction, and therefore the size of the peer effect,

may differ by course type.

Adjusting equation (16) to incorporate abilities and peer influence that vary by course15Information as to which courses were assigned to which course types is available from the authors upon

request.

16

type yields:

Yikt = αik +N∑

j 6=i

Iijt

(γαjk

Mit

)+

C∑

c=1

Iictδc + εikt (17)

where k indexes course type. This model is run through our algorithm separately for each type

of course, yielding three sets of peer and class effects estimates, as well as separate student

ability measures for each course type taken. Note that while classes with only one section do

not help in the identification of γ directly, these classes are still useful in identifying the α’s.

Table 4 shows the results from estimating equation (16) for each of the three types of

courses. The results indicate positive and significant section peer effects for all course types.

The magnitudes of the section-level peer effects suggest that peer effects are most important

in the social sciences and least important in math and science. This pattern may reflect the

amount of collaborative work required in each course type as well as the differing amounts of

discussion that occur in the sections.

In order to understand the importance of peer ability relative to own ability, we need to

take into account the differences in variation of peer and own ability. There is likely to be

less variation in peer ability than in own ability as peer ability averages over a cross-section

of students leading to some heterogeneity canceling out. The first two columns of Table 5

show the standard deviation of mean peer ability and the standard deviation of individual

ability, respectively. The third column then shows the fraction of a standard deviation of own

ability that is equivalent, in terms of its effect on grades, to a one-standard-deviation increase

in peer ability. This is calculated by dividing the standard deviation of mean peer ability

by the standard deviation of individual ability and multiplying this number by the estimated

gamma.

The gap evident in the raw marginal effects between math and science and the other

course types is somewhat mitigated because there is relatively more heterogeneity in peer

ability in math and science courses than in humanities or social science courses. A one-

standard-deviation increase in peer ability is shown to be equivalent to a maximum of 9% of

the effect of a one-standard-deviation increase in individual ability in the social sciences, and

to a minimum of 3% of the effect of a one standard deviation increase in individual ability in

math and science courses.

17

5.2 Heterogeneous Gamma Model

The assumption that each student is affected in the same manner by their classmates is restric-

tive and not as interesting from a policy perspective. In particular, the linear-in-means model

implies that, in terms of grades, any winners from reshuffling peers are perfectly balanced

by those who lose from the reshuffling.16 To relax this assumption, we apply our iterative

estimator to an enhanced spillover model in which peer effects are allowed to vary with the

observable characteristics of the student:

Yikt = αik +N∑

j 6=i

Iijtαjk

Mit(Xiγ) +

C∑

c=1

Iictδc + εikt (18)

where X includes whether the student is female, whether the student is Asian or another race,

as well as the student’s SAT math and SAT verbal scores.

Table 6 shows the results from this heterogeneous peer effects model. The qualitative

results for humanities and social sciences are the same. Relative to white males, Asians see

less of a return to peer ability while females and other non-white students see higher returns.

Both SAT math and SAT verbal scores are associated with higher returns to peer ability.

While Asians in math and science again see lower returns to peer ability, females and other

non-whites also see lower returns relative to their white male counterparts. The interaction

of the peer effect with SAT verbal score is once again positive, but the sign on the SAT math

interaction is now negative.

The differences in the SAT interactions across fields suggest that two competing forces

may be at play. First, those with higher test scores may have skills that make them better

able to benefit from their peers. Working against this, however, is that there is more scope

for students to benefit the lower they are in the ability distribution. SAT verbal and math

scores may not be highly correlated with the ability to perform well in the humanities and

social sciences implying that the first effect dominates in these course types. However, the

SAT math score may be highly correlated with the ability to perform well in math and science

classes, leading to the second effect dominating.

Averaging across all individuals within a course type, the relative magnitude of the peer

effects is unchanged from the homogenous gamma model. Peer ability is most important in16See Hoxby and Weignarth (2005) for a discussion.

18

social science courses and least important in math and science courses. However, the overall

magnitude of the peer effects is significantly higher, increasing by an average of over 40%

across course type. Relative to a one-standard-deviation increase in own ability, the effects of

a one-standard-deviation increase in peer ability are also higher in the heterogeneous gamma

model as shown in the fourth column of Table 7. The ratio of the effects of a one-standard-

deviation increase in mean peer ability to a one-standard-deviation increase in own ability

range from a low of 3.5% for math and science to a high of 10.5% in the social sciences.

5.3 Analysis of Ability

Next, we explore the extent to which the fixed effects from our iterative algorithm are pre-

dictable using the observed proxies for ability consistently used in related literature, and to

what extent the fixed effects estimated using the two methods differ. In order to facilitate

this comparison, we use SAT scores, high school performance, and a host of other observable

student attributes as regressors to construct a conglomerate observable measure of ability.

This approach is analogous to the creation of an academic index (as employed by Sacerdote

(2001)). For each course type, we regress our estimated student fixed effects on an array of

previous performance measures and demographics. These results are presented Table 8.

Columns 1 through 3 of Table 8 show results when we use the fixed effects estimated in

our homogeneous gamma model as the dependent variable. The statistical significance and

magnitudes of the coefficients on SAT math and SAT verbal scores vary across the four dif-

ferent course types in predictable ways. For example, SAT math scores are insignificant when

explaining ability in humanities courses, but are a better proxy for math and science ability.

The opposite is true for SAT verbal scores, with higher SAT verbal scores associated with

higher ability in the humanities but uncorrelated with ability in math and science. Consistent

with Arcidiacono (2004), females outperform males across all course types. Conditional on

SAT scores and high school GPA, whites perform better than all other ethnic groups including

Asians.

The second set of columns in Table 8 shows the corresponding results for the heterogeneous

gamma model. Recalling the results found in Table 6 regarding the positive association of

SAT verbal scores with stronger peer effects across all course types, it is unsurprising that the

coefficient on own SAT verbal score is smaller and even negative for some course types when

19

predicting own ability. Previous work such as Arcidiacono (2004), Arcidiacono and Vigdor

(2007), and Arcidiacono et al. (forthcoming) have all found no returns to verbal test scores in

the labor market. Combining these findings with the results presented here suggests that SAT

verbal scores have very little to with ability in the absolute, but rather reflect how capable an

individual is at extracting rents from others.

We find that the R2 for these regressions ranges from 0.13 to 0.36 depending on the course

type and whether we use the homogeneous or heterogeneous gamma model to generate the

individual fixed effects. That these observable characteristics only explain a small portion

of our estimated ability measures suggests the possibility of large biases associated with the

unobserved ability problem when following a selection-on-observables, or random assignment,

approach. With much of peer ability unobserved, we would expect to underestimate the extent

to which outcomes are affected by one’s peers. However, selection into sections may occur

based upon one’s own unobserved ability, which in turn may be correlated with observed peer

ability. This latter effect characterizes the selection problem, and is likely to bias peer effects

estimates upward. While random assignment circumvents this latter problem, a downward

bias remains from not taking into account the impact of unobservable peer attributes on

outcomes.

6 Quantifying Selection

In this section we quantify how much selection is taking place within course types with respect

to both observed and unobserved ability.17 For ease of exposition, we refer to the predicted

values of the regressions in Table 8 as ‘observed ability’ and the residuals of those regressions as

‘unobserved ability’. Thus, by construction, observed and unobserved ability are uncorrelated

at the individual level. However, unobserved individual ability and observed peer ability

will be correlated if students sort by total ability. By decomposing ability into its observed

and unobserved components, we can calculate the correlation between unobserved individual

ability and observed peer ability which is the crux of the selection problem. We also examine

selection across course types. In particular, we can split our student sample by which course

type was chosen the most. Because we estimate separate abilities for each course type, we17See Altonji et al. (2005) for more discussion of selection on observed and unobserved factors.

20

are able to determine whether students choose to take more courses in areas where they are

comparatively more able. In addition to examining comparative advantage, we can also see

whether students who choose particular course types enjoy an absolute performance advantage

by determining if their mean fixed effects are higher on average across all course types.

6.1 Selection Within Course Types

Table 9 provides information by course type on the selection evident with respect to both ob-

served and unobserved ability. We use the underlying ability as estimated by the homogeneous

gamma model and the heterogeneous gamma model, as well as the observed and unobserved

portions of this ability. The first row for every course type shows the section-size-weighted

average of the section-wide standard deviation of the variable in question, across all sections

in the particular course type; the second row for every course type shows the simple standard

deviation of the variable in question across the sample of students taking courses of the given

course type. The third row provides the ratio of the two. The smaller the numbers in the

third row, the tighter is the distribution of the variable within sections relative to the unsorted

distribution, and therefore, the more selection is evident with respect to that variable.

This table allows a direct analysis of the comparative degree of selection evident with

respect to observed versus unobserved ability. The patterns for both the homogeneous speci-

fication and the heterogeneous specification are almost exactly the same. For all three course

types there is more selection on unobserved ability than on observed ability, with higher ratios

of section standard deviations to population standard deviations found for observed ability. In

the social sciences and in particular for the humanities there is more selection on the estimated

α’s as a whole than on either observed ability or unobserved ability separately, with the high-

est levels of selection found in math and science. These patterns are driven by the correlation

between peer observed and unobserved ability. For the homogeneous gamma specification, the

correlation coefficients between peer observed and unobserved ability are 0.03, 0.07, and 0.35

in the humanities, social sciences, and math and science, respectively. The selection on the

estimated α’s in math and science is particularly strong relative to selection on either observed

or unobserved ability, which is consistent with the high correlation between peer observed and

unobserved ability.

With the observed and unobserved ability measures it is also possible to estimate the

21

correlation between unobserved individual ability and observed peer ability, which feeds di-

rectly into the bias associated with the selection problem. The correlation coefficients for

unobserved individual ability and observed peer ability are 0.03, 0.06, and 0.20 for humani-

ties, social sciences, and math and science, respectively. The high correlation coefficient for

math and science suggests that the upward bias associated with peer effect estimation using a

selection on observables approach might be quite large. We test this hypothesis in section 7.

6.2 Selection Across Course Types

Because the vast majority of students are observed in courses of multiple types during their

tenure at Maryland, we obtain multiple estimates of ability for most students. Each estimated

ability measure is specific to courses of a particular type, by virtue of the estimation procedure,

and as such each can be assumed to reflect a skill set that is particularly in demand in that

course type. Calculating the correlations between estimated ability levels illuminates the

extent to which good performance in each of the three course types is driven by similar student

attributes as performance in the other course types, and therefore provides an empirical index

of the academic similarity of course types. For ease of exposition, we focus on the homogeneous

gamma model for the rest of the paper.

Panel A of Table 10 shows the correlation coefficients among estimated ability levels across

the three course types from the homogeneous gamma model. These correlations are created

using estimated abilities from students observed in all course types.18 The correlation coeffi-

cients are all quite large and close together, ranging from 0.65 to 0.69.19 Panel B of Table 10

displays analogous results using only the portion of estimated abilities for each student that

is predictable using our observable variables. The relative relationships amongst observable

abilities by course type are the same, with observable abilities to perform in the humanities

and in math and science being related to a lesser extent than those of the other two course type

pairings (humanities and social sciences, and social sciences and math and science). However,

the strength of the relationships is much stronger across the board with correlation coefficients18Correlations among estimated abilities were also calculated for all students who were observed in each pair

of course types. Similar coefficients resulted.19The pattern of correlations for the heterogenous gamma model is similar, although the correlation coeffi-

cients are slightly lower: 0.64 for social sciences and humanities, 0.62 for social sciences and math and science,

and 0.54 for math and science and humanities.

22

above 0.95 for observed abilities for two out of three pairings. The differences across the pan-

els suggest that ability is much more heterogeneous than can be captured by observed ability

measures.

With this information as background, Table 11 illustrates the degree to which individual

students are observed to be sorting into the types of courses for which they appear, based on

our model, to be best suited. In particular, we label a student as specializing in a particular

course type if the number of courses taken in that course type is higher than the number of

courses taken in either of the other two course types. Panel A of this table displays results

using the estimated abilities from our homogeneous gamma model standardized to a N(0,1)

distribution, and Panel B displays results using only the portion of those estimated abilities

that could be predicted based on observable characteristics, also standardized to N(0,1). The

rows correspond to the sets of students who specialize in humanities, social sciences, and math

and science, respectively. The numbers along the rows gives the mean (normalized) fixed effect

for each of the different course types.

We see two striking patterns in Panel A. First, students who specialize in humanities

courses are estimated to be less able across the board than those who specialize in either

the social sciences or math and science. Students specializing in math and science are the

most able across the board, with higher average fixed effects in each course type. Even more

interesting, students in each specialization group appear to have specialized in the area for

which they are most suited. This can be seen from the fact that the highest number in each row

is that corresponding to the course type specialized in by that sample of students–the highest

number in each row is on the diagonal.20 Panel B of Table 11, where we use only observed

ability to examine selection, also illustrates both absolute and comparative advantage, but

reflects more attenuated distributions of abilities and less heterogeneity than shown in Panel

A.20Paglin and Rufolo (1990) find similar sorting patterns using GRE exam data and transcript data from the

University of Oregon and Oregon State University. They find that students with high math ability tend to take

courses in which there is a high return to this skill.

23

7 Comparing the Method to Conventional Methods

To compare our estimated peer effects to those that would be obtained using conventional

techniques, we first conceptualize the estimation problem as follows. Given a model where the

ability of each student can be decomposed into observed versus unobserved portions, there

are two econometric obstacles to the accurate estimation of the spillover. The first obstacle

is a positive correlation between the student’s own unobserved ability and his peer group’s

observed ability, which also leads in any given sample to a correlation between the peer group’s

observed ability and the peer group’s unobserved ability. This problem leads to an upward

bias of the spillover parameter. The second obstacle is that when only observables are used

to form the peer ability measures, the underlying distribution of peer ability is attenuated,

leading to downward pressure on the estimated impact of a one-standard-deviation change in

peer ability.

To examine the quantitative impact of these two problems separately, we first artificially

eliminate the correlation between the student’s own unobserved ability and the peer group’s

observed ability, by differencing out our estimated individual fixed effects and course effects

from student grades. We regress these adjusted grades on observed peer ability, rather than

total peer ability. This enables us to examine the consequences for estimation of using an

incomplete measure of peer ability in a case where the link between individual unobservables

and peer observables is broken.

Row 2 of Table 12 for each course type presents the results of this first exercise, where

we use total ability (our estimated α’s) for the individual and observed ability for peers.

For comparison, Row 1 of the table for each course type gives the original spillover estimate

produced using our algorithm. Looking at Column 3 of the table, we can see that the estimated

effects of a one-standard-deviation increase in observed peer ability are at most two-thirds the

size of a one-standard-deviation increase in total peer ability. This first-order decrease in effect

magnitude is evident because the impact of unobserved peer ability is not captured in Row 2

except through the correlation between observed and unobserved peer ability.

However, under random assignment, a one-standard-deviation increase in peer ability

would produce even smaller effects than those shown in Row 2 of Table 12 for each course

type. There are two reasons for this. First, peer observed ability and peer unobserved ability

24

are positively correlated in our data, but would not be correlated under random assignment.21

This leads to an upward bias on our estimates of the spillover parameters: the estimated

coefficients in the second row for each course type are all higher than those in the first row.

The largest percentage increase in the peer ability coefficient is for courses in math and sci-

ence, where the correlation between observed and unobserved peer ability is the highest. The

second reason why random assignment will lead to even lower estimates of a one-standard-

deviation increase in peer ability is that random assignment itself leads to less heterogeneity

in mean peer ability across sections when higher ability students choose sections with other

high ability students. This attenuation in the distribution of peer ability means that a one-

standard-deviation increase in peer ability will be smaller under random assignment than in

a self-selected context.

We next investigate what happens when the econometrician additionally assumes that

students select into peer groups only based on observable characteristics (the “selection on

observables” approach). In Row 3 of Table 12, we again use observed peer ability, but now

we use observed ability for the individuals as well rather than the estimated fixed effects. The

positive correlation between unobserved individual ability and observed peer ability biases the

selection-on-observables estimate of the spillover parameter upward. While the estimate of the

spillover parameter is biased upward, the effect of a one-standard-deviation increase in peer

ability may still be smaller because the variance in observed peer ability is smaller than the

variance in peer ability as a whole. As can be seen in the third column, this is indeed the case

for humanities, the course type with the smallest correlation between unobserved individual

ability and observed peer ability. For the social sciences, the selection-on-observables estimate

of a one-standard-deviation increase, although higher than our original estimate, is still closer

than the estimates given in the second row that mitigate the selection problem. However,

for math and science the estimated effect of a one-standard-deviation increase in peer ability

is significantly higher using the selection-on-observables approach than using our algorithm.

This is driven by (1) the high correlation between unobserved individual ability and observed

peer ability in math and science; (2) the fact that observed ability is a greater fraction of21Recall that observed and unobserved ability are uncorrelated by construction at the individual level. Under

random assignment, observed and unobserved peer ability will also be uncorrelated. The positive correlation

in observed and unobserved peer ability in our data stems from student sorting based upon total ability.

25

total ability in math and science; and (3) the fact that the underlying peer effect estimate

from our method is smallest in math and science, which mitigates the underestimation of a

one-standard-deviation increase in peer ability.22

8 Conclusion

Accurate estimation of peer effects in the classroom is plagued by at least two issues, both

of which have to do with ability not being fully observed. First, there is selection into the

peer group which leads to a positive correlation between unobserved individual ability and

observed peer ability. If ignored, this correlation leads to biased upward estimates of peer

effects parameters. On the other hand, underestimation of the effects of peers may result from

ignoring peer effects through unobservables.

We present a new iterative method for estimating educational peer effects that overcomes

both these obstacles. All that is required is that there are multiple observations per student

with the peer group changing over time. We then control for individual effects and allow the

peer effect to operate through a linear combination of the other individual fixed effects. We

show that our estimator is consistent and asymptotically normal for fixed T as N goes to

infinity. We also develop an iterative algorithm that is computationally much cheaper than

direct non-linear least squares minimization yet produces the non-linear squares results upon

convergence. Monte Carlo results suggest that the model performs quite well even when the

number of observations per student is small.

We estimate the model on transcript data from the University of Maryland. Small but

significant peer effects are found, with evidence of heterogeneity by course type. Social science

courses show the largest peer effects, whereas grades in math and science courses rely least on

peer ability and most heavily on a student’s own ability.

Previous efforts to estimate spillover effects in education that do not rely on random

assignment are often plagued by concerns regarding selection on unobservables. The data

suggest that this is a valid issue. Students select into sections based more on unobservable

factors than on observable factors. This leads to correlation in unobserved own ability and

observed and unobserved peer ability that, if ignored, biases the spillover parameter upward.22Indeed, if the spillover parameter were zero, there would be scope for the attenuation effect.

26

There is also much selection across course type. Students sort into course types where their

relative abilities are highest, suggesting comparative advantage is important in the selection

of courses. However, absolute advantage is also present as those who primarily choose math

and science course are more able in both humanities and social sciences than those who choose

to specialize in one of the other ares.

Our method allows us to quantify the effects of both the selection problem and the prob-

lem of not being able to estimate peer effects through unobservables. Our different course

types illustrate how the setting dictates which of these problems is more important. For

math and science courses, the estimated spillover parameter from our model is small. This,

coupled with much selection into math and science courses, leads to estimates from a selection-

on-observables approach that significantly overstates the importance of peers. However, for

humanities courses the estimated spillover parameter is larger than in math and science. Cou-

pled with much less selection than in math and science, the selection-on-observables approach

places similar importance on peers to what is estimated from our model. This is in contrast

to random assignment, which removes the correlation between individual unobserved ability

and peer observed ability but ignores peer effects through unobservables, leading to random

assignment significantly understating the impact of peers on achievement.

There are many avenues to be explored in future research. First, future research will relax

the assumption that an individual’s ability to help others is proportional to an individual’s

own ability to perform well. The individual who asks the clarifying question may be more

useful to others than a smarter individual who remains quiet. It is possible to extend the

model such that spillover ability is treated differently from the ability to perform well for

oneself.

Second, peer effects here are purely transitory. Future work will relax this assumption

by allowing the effect of peer ability in a particular course to decay over time, as well as

to influence performance in other contemporaneous classes. This will help us determine the

long-run impact of peer groups on educational achievement, and may result in higher peer

effect estimates as we account for spillovers beyond the classroom.

Finally, rather than separately estimating ability in each course type, the model could be

expanded to allow a factor structure on ability and allow the returns to the different abilities

to vary by course type. This expansion would allow a better exploitation of large data sets,

27

such as ours, with outcomes that have heterogeneous determinants, since performance on all

outcome measures collectively would be used to estimate individual abilities.

28

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sity of Chicago Press.

Woutersen, T.: 2002, “Robustness Against Incidental Parameters”. Unpublished Manuscript.

Zimmerman, D. J.: 2003, “Peer Effects in Higher Education: Evidence from a Natural Exper-

iment”, Review of Economics and Statistics 85(1), 9–23.

31

Table 1: Monte Carlo Simulations: True Gamma = 0Obs. Per Peer Group Random Assignment Selection

Student Size σε=1.95 σε=1.15 σε=1.95 σε=1.15

2 2 γ 0.0011 -0.0002 -0.0094 -0.0042

(0.035) (0.015) (0.060) (0.023)

R2 0.7044 0.8209 0.7055 0.82

5 10 γ 0.0005 0.0011 0.0039 -0.0055

(0.037) (0.021) (0.061) (0.032)

R2 0.4831 0.6845 0.4815 0.6852

10 10 γ 0.0049 -0.0018 0.0013 -0.0003

(0.026) (0.014) (0.036) (0.015)

R2 0.415 0.644 0.4148 0.6453

Note: The R-squared values reported in this table are those pertaining to the

regression of grades onto the constructed fixed effect values. We alter the random

error added on to the constructed grade for each student in order to manipulate

the amount of variation in performance that is explained by the ability measure.

66

Table 2: Monte Carlo Simulations: True Gamma = .15Obs. Per Peer Group Random Assignment Selection

Student Size σε=1.95 σε=1.15 σε=1.95 σε=1.15

2 2 γ 0.1507 0.1511 0.14 0.1508

(0.034) (0.016) (0.060) (0.024)

R2 0.7055 0.8219 0.7125 0.8277

5 10 γ 0.1499 0.1498 0.1463 0.1489

(0.041) (0.021) (0.059) (0.033)

R2 0.4822 0.6862 0.4943 0.6967

10 10 γ 0.1502 0.148 0.148 0.1524

(0.025) (0.012) (0.036) (0.018)

R2 0.4152 0.644 0.4288 0.6594

Note: The R-squared values reported in this table are those pertaining to the

regression of grades onto the constructed fixed effect values. We alter the random

error added on to the constructed grade for each student in order to manipulate

the amount of variation in performance that is explained by the ability measure.

67

Tab

le3:

Sam

ple

Size

s

S99

F99

S00

F00

S01

F01

TO

TA

L

(1)

Stud

ent-

Sect

ions

27,9

0037

,231

37,1

0945

,991

45,0

5453

,546

246,

831

(2)

Stud

ents

7126

9646

9458

11,7

6011

,393

13,6

6263

,045

(3)

Uni

que

Sect

ions

3095

3388

3408

3628

3543

3754

20,8

16

(4)

Uni

que

Cou

rses

1030

1079

1172

1189

1246

1252

6968

(5)

Sing

le-S

ecti

onC

ours

es63

268

273

375

277

879

543

72

(6)

Stud

ent-

Sect

ions

23,2

0631

,627

30,5

9938

,056

36,2

6043

,853

203,

601

(Onl

yM

ulti

-Sec

tion

Cou

rses

)

Not

e:Fig

ures

repr

esen

tth

eda

tase

taf

ter

appl

ying

the

rest

rict

ions

note

din

the

text

.T

he

unre

stri

cted

data

set

cont

aine

d35

1,94

0st

uden

t-se

ctio

nob

serv

atio

ns.

Row

s3

and

4sh

owth

e

tota

lnum

berof

uniq

uese

ctio

nsan

dco

urse

s,re

spec

tive

ly,i

nw

hich

anyo

nein

the

sam

ple

duri

ng

the

give

nse

mes

ter

was

obse

rved

.

68

Table 4: Peer Effects Results by Course Type: Homogeneous Gamma Model

Humanities Soc.Sci. Math/Sci.

Section peer ability 0.1613 0.1960 0.0483

(0.0007) (0.0008) (0.008)

N 86,844 77,312 82,675

R2 0.6373 0.6321 0.6861

Note: Dependent variable is the grade in the class. Class

fixed effects are estimated in all specifications.

69

Tab

le5:

Stan

dard

Dev

iati

ons

ofE

stim

ated

Abi

lity

and

Mar

gina

lE

ffect

s:H

omog

eneo

usG

amm

aM

odel

Cou

rse

Typ

eSe

ctio

nSt

dD

evPop

ulat

ion

Std

Dev

Mar

gina

lE

ffect

Rat

io

Hum

anit

ies

0.28

230.

6804

0.06

69

Soci

alSc

ienc

e0.

3103

0.71

250.

0853

Mat

han

dSc

ienc

e0.

5952

0.94

980.

0302

Not

e:“S

ecti

onSt

dD

ev”

isth

est

anda

rdde

viat

ion

ofav

erag

epe

erab

ility

acro

ssth

esa

mpl

e

ofst

uden

t-se

ctio

nob

serv

atio

nsof

the

give

nco

urse

type

.“P

opul

atio

nSt

dD

ev”

isth

e

stan

dard

devi

atio

nof

abili

tyac

ross

the

sam

ple

ofst

uden

t-se

ctio

nob

serv

atio

nsin

cour

ses

of

the

give

nco

urse

type

.T

hese

calc

ulat

ions

are

both

base

don

the

fixed

effec

tses

tim

ated

by

the

hom

ogen

eous

gam

ma

mod

el.

“Mar

gina

leff

ect

rati

o”sh

ows

the

rati

oof

(1)

the

effec

t

ongr

ades

from

aon

e-st

anda

rd-d

evia

tion

incr

ease

inpe

erab

ility

to(2

)th

eeff

ect

ongr

ades

from

aon

e-st

anda

rd-d

evia

tion

incr

ease

inow

nab

ility

.

70

Table 6: Peer Effects Results by Course Type: Heterogeneous Gamma Model

Humanities Soc.Sci. Math/Sci.

Section peer ability 0.2058 0.2227 0.0940

(0.0008) (0.0013) (0.0013)

Section peer ability*Female 0.0970 0.0584 -0.0517

(0.0010) (0.0018) (0.0018)

Section peer ability*Asian -0.0098 -0.0347 -0.0346

(0.0016) (0.0026) (0.0022)

Section peer ability*Other Nonwhite 0.0375 0.0252 -0.0420

(0.0014) (0.0026) (0.0026)

Section peer ability*SATm 0.0410 0.0507 -0.0560

(0.0006) (0.0012) (0.0011)

Section peer ability*SATv 0.0222 0.0147 0.0635

(0.0005) (0.0010) (0.0009)

N 86,844 77,312 82,675

R2 0.6376 0.6323 0.6864

Note: Dependent variable is the grade in the class. Class fixed effects are

estimated in all specifications.

71

Tab

le7:

Stan

dard

Dev

iati

ons

ofE

stim

ated

Abi

lity

and

Mar

gina

lE

ffect

s:H

eter

ogen

eous

Gam

ma

Mod

el

Cou

rse

Typ

eSe

ctio

nSt

dD

evPop

ulat

ion

Std

Dev

Avg

Mar

gina

lE

ffect

Mar

gina

lE

ffect

Rat

io

Hum

anit

ies

0.23

830.

6368

0.25

950.

0970

(.06

22)

Soci

alSc

ienc

e0.

2849

0.67

470.

2480

0.10

48

(.05

54)

Mat

han

dSc

ienc

e0.

6028

0.96

600.

0555

0.03

47

(.06

36)

Not

e:“S

ecti

onSt

dD

ev”

isth

est

anda

rdde

viat

ion

ofav

erag

epe

erab

ility

acro

ssth

esa

mpl

eof

stud

ent-

sect

ion

obse

rvat

ions

ofth

egi

ven

cour

sety

pe.

“Pop

ulat

ion

Std

Dev

”is

the

stan

dard

devi

atio

nof

abili

ty

acro

ssth

esa

mpl

eof

stud

ent-

sect

ion

obse

rvat

ions

inco

urse

sof

the

give

nco

urse

type

.T

hese

calc

ulat

ions

are

both

base

don

the

fixed

effec

tses

tim

ated

byth

ehe

tero

gene

ous

gam

ma

mod

el.

Col

umn

3sh

ows

the

mar

gina

leffe

cton

grad

esfr

oma

one-

poin

tin

crea

sein

peer

abili

ty.

“Mar

gina

leffe

ctra

tio”

show

sth

era

tio

of(1

)th

eeff

ect

ongr

ades

from

aon

e-st

anda

rd-d

evia

tion

incr

ease

inpe

erab

ility

to(2

)th

eeff

ect

ongr

ades

from

aon

e-st

anda

rd-d

evia

tion

incr

ease

inow

nab

ility

.T

henu

mbe

rsin

pare

nthe

ses

are

stan

dard

devi

atio

ns

ofth

em

argi

naleff

ects

ofa

one-

poin

tin

crea

sein

peer

abili

tyth

atar

ees

tim

ated

tooc

cur

inth

esa

mpl

e.

72

Table 8: Regression of Fixed Effects on Observed Ability

Homogeneous Gamma Model Heterogenous Gamma Model

Hum. Soc.Sci. Math/Sci. Hum. Soc.Sci. Math/Sci.

SATm 0.00 0.11 0.36 -0.20 -0.04 0.51

(0.01) (0.01) (0.01) (0.01) (0.01) (0.01)

SATv 0.08 0.11 0.00 -0.04 0.06 -0.19

(0.01) (0.01) (0.01) (0.01) (0.01) (0.01)

HS GPA 0.49 0.49 0.73 0.49 0.49 0.72

(0.01) (0.02) (0.02) (0.01) (0.02) (0.02)

Female 0.26 0.17 0.15 -0.15 0.03 0.27

(0.01) (0.01) (0.01) (0.01) (0.01) (0.01)

Black -0.28 -0.23 -0.18 -0.44 -0.29 -0.10

(0.02) (0.02) (0.02) (0.02) (0.02) (0.02)

Hispanic -0.15 -0.19 -0.17 -0.31 -0.25 -0.08

(0.03) (0.04) (0.04) (0.03) (0.04) (0.04)

Asian -0.12 -0.13 -0.10 -0.08 -0.05 -0.02

(0.02) (0.02) (0.02) (0.02) (0.02) (0.02)

Amer. Ind. -0.39 -0.35 -0.31 -0.55 -0.41 -0.22

(0.12) (0.12) (0.16) (0.12) (0.12) (0.16)

Honors 0.14 0.15 0.17 0.14 0.15 0.17

(0.02) (0.02) (0.02) (0.02) (0.02) (0.02)

Sports -0.07 -0.14 0.01 -0.08 -0.14 0.00

(0.02) (0.03) (0.03) (0.02) (0.03) (0.03)

In-state 0.05 0.08 0.12 0.05 0.08 0.12

(0.03) (0.03) (0.03) (0.03) (0.03) (0.04)

N 17,332 15,264 16,077 17,332 15,264 16,077

R2 0.22 0.24 0.34 0.13 0.17 0.36

Note: The dependent variable in columns 1 through 3 is the student-level

fixed effects estimated in the homogeneous gamma model; the dependent

variable in columns 4 through 6 is the student-level fixed effects estimated

in the heterogeneous gamma model. The excluded racial/ethnic category is

white; racial/ethnic categories are mutually exclusive. Standard errors are

robust to heteroskedasticity.

73

Tab

le9:

Sele

ctio

nba

sed

onO

bser

vabl

esan

dE

stim

ated

Abi

lity

Hom

ogen

eous

Gam

ma

Het

erog

eneo

usG

amm

a

Cou

rse

Typ

eα

α̂α

uα

α̂α

u

Hum

anit

ies

Avg

Sect

ion-

leve

lSD

.615

4.3

372

.538

5.5

849

.248

7.5

415

Pop

ulat

ion

SD.8

046

.377

3.7

106

.761

9.2

711

.712

1

Rat

io.7

649

.893

7.7

578

.767

7.9

174

.760

4

Soci

alSc

ienc

e

Avg

Sect

ion-

leve

lSD

.636

5.3

750

.566

7.6

056

.299

4.5

688

Pop

ulat

ion

SD.8

618

.422

8.7

510

.822

6.3

343

.751

7

Rat

io.7

386

.886

9.7

546

.736

2.8

956

.756

7

Mat

han

dSc

ienc

e

Avg

Sect

ion-

leve

lSD

.747

5.5

036

.666

6.7

636

.532

3.6

649

Pop

ulat

ion

SD1.

0567

.616

5.8

583

1.07

19.6

432

.857

5

Rat

io.7

074

.816

9.7

767

.712

4.8

276

.775

4

Not

e:E

ach

set

ofro

ws

corr

espo

nds

tose

ctio

nsin

one

cour

sety

pe;

each

set

of

colu

mns

corr

espo

nds

toon

eve

rsio

nof

the

mod

el(h

omog

eneo

usga

mm

ave

rsus

hete

roge

neou

sga

mm

a).

Var

iabl

esun

der

anal

ysis

appe

arin

the

head

ing

row

:α

isab

ility

ases

tim

ated

byou

rm

odel

,α̂

isob

serv

edab

ility

,an

dα

uis

unob

serv

ed

abili

ty.

“Avg

Sect

ion-

leve

lSD

”is

the

aver

age

(acr

oss

alls

ecti

ons,

and

wei

ghte

dby

sect

ion

size

)of

the

stan

dard

devi

atio

nof

the

vari

able

wit

hin

ase

ctio

n.“P

opul

atio

n

SD”

isth

est

anda

rdde

viat

ion

ofth

eva

riab

lein

the

popu

lati

onof

stud

ents

taki

ng

cour

ses

ofth

egi

ven

cour

sety

pe.

“Rat

io”

isth

era

tio

ofth

efir

stof

thes

eto

the

seco

nd,a

ndsh

owsth

ede

gree

ofse

lect

ion

into

sect

ions

wit

hre

spec

tto

each

vari

able

disp

laye

d.

74

Table 10: Correlations of Estimated Abilities Across Course Types

Panel A: Abilities

Course type Humanities Social Science

Humanities 1.0000

Social Science 0.6875 1.0000

Math and Science 0.6469 0.6776

Panel B: Predicted abilities

Course type Humanities Social Science

Humanities 1.0000

Social Science 0.9643 1.0000

Math and Science 0.8808 0.9593

The abilities used in these correlation matrices are those of

the 12,715 students who took classes in all three course types,

and they are estimated by the homogeneous gamma model.

Panel A displays correlations amongst the full abilities, while

Panel B displays correlations amongst the predicted values

from regressing the estimated abilities from the homogeneous

gamma model on observable variables (as shown in the first

three columns of Table 8).

75

Table 11: Specialization of Students into Course Types By Relative Aptitude

Panel A: Abilities

Humanities Soc.Sci. Math/Sci. N

Humanities-specializers -.08 -.19 -.26 3978

Social Science-specializers .04 .10 -.07 3547

Math and Science-specializers .14 .21 .43 3745

Panel B: Predicted abilities

Humanities Soc.Sci. Math/Sci. N

Humanities-specializers -.04 -.13 -.21 3977

Social Science-specializers -.11 -.10 -.11 3547

Math and Science-specializers .18 .27 .36 3745

In Panel A, each cell shows the mean of the deviations of students’ ability to

perform in the course type of that column (as estimated by our homogeneous

gamma model, and standardized to a normal (0,1) distribution) from the

sample standardized mean of estimated ability across the course type, for

the population of that row. “Specializers” are students who are observed to

take more courses in the given course type than in either of the other two

course types.

76

Table 12: Comparing the Method to Conventional Results: Homogeneous Gamma Model

Own Section peers’ Effect of 1-stddev Own Peers’

ability ability chg in peer ability ability ability

Humanities 1 0.1613 0.0455 Total Total

(–) (0.0007)

1 0.1642 0.0285 Total Observed

(–) (0.0008)

0.9349 0.2502 0.0435 Observed Observed

(0.0076) (0.0077)

Social Science 1 0.1960 0.0608 Total Total

(–) (0.0008)

1 0.2061 0.0370 Total Observed

(–) (0.0009)

0.8518 0.4085 0.0733 Observed Observed

(0.0077) (0.0077)

Math and Science 1 0.0483 0.0287 Total Total

(–) (0.0008)

1 0.0516 0.0193 Total Observed

(–) (.0009)

0.7784 0.3519 0.1313 Observed Observed

(0.0062) (0.0063)

Note: Dependent variable is grade in the class. For models shown in Rows 1 and 2 for each

coursetype, class effects are removed before estimation by demeaning using the ‘true’ values

of the class effect as estimated by our homogeneous gamma model. For Row 3, class fixed

effects are absorbed in estimation. For the purposes of this table, we ignore the sampling

variation in the parameter estimates used to construct our observed ability measures, which

may impact the standard errors reported here. Observations are as in Table 4, although

creating the second and third rows involved dropping observations for which observables

were missing. The maximum number of observations dropped for a course type was 6.

77